{"id":"4f9b73b9-09e7-49b1-8b23-41c2656c9310","arxiv_id":"2411.13675","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"In Milgrom-law dynamics, a simulated low-mass open star cluster develops a leading tidal tail with up to twice as many stars as the trailing tail and dissolves about 25% faster than in Newtonian gravity.","lead":"This paper simulates 400-star open clusters using a version of Milgrom's modified gravity law, called Milgrom-law dynamics, and finds that many more stars end up in the leading tidal tail than in the trailing one. The clusters also lose stars about 25% faster than in standard Newtonian gravity, offering a small-scale test for MOND theories.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Softening ε=0.001 pc is ~24× too small to suppress Newtonisation (transition at r≈0.024 pc for 0.5 Msun); transient Newtonian binaries contaminate the cluster, so the tail asymmetry and dissolution rate may depend on an ad hoc parameter.","rationale":"The reader identifies the MLD postulate (Eq. 10) and the softening as the weakest assumption; I agree, and I sharpen the concern. The paper's own Sec. 4.2 shows that internally Newtonian binaries follow Newtonian galactic orbits in MLD, and the cluster models are intended to avoid this by setting ε=0.001 pc. However, that softening length is roughly 24 times smaller than the separation at which a 0.5 Msun pair reaches a0 (r≈0.024 pc), so pairs in the range (0.001, 0.024) pc are common during relaxation and behave Newtonianly. The stated purpose of the softening is therefore not met, and the simulation contains Newtonian contamination whose influence on the tidal-tail asymmetry and evaporation rate is untested. The proposed ε=0.03 pc run directly isolates this parameter: it removes all internally Newtonian pairs while preserving the MOND external-field-effect regime of the cluster, providing a decisive check on whether the headline q-parameter and dissolution ratio are robust. Until such a test is performed, the Sec. 6 conclusion that the asymmetry is a property of 'the general MONDian dynamical concept' is not established by this work. This does not change the reader's CONDITIONAL verdict, but it adds a specific, actionable condition: demonstrate robustness to the softening choice and report the close-encounter statistics. The paper is otherwise internally consistent in its equations and numerical implementation, and it is honest about MLD being an approximation, so no stronger verdict is warranted.","tokens_in":19246,"tokens_out":15105,"duration_ms":1048413,"concrete_test":"Rerun the five MOND cluster models with ε increased to 0.03 pc — the value at which the softened maximum two-body acceleration (≈0.385 Gm/ε²) falls below a0 for a 0.5 Msun pair, so no transient pair is internally Newtonian — keeping all other initial conditions identical. Compare the time evolution of q50-200 and Ncl/N0 with the ε=0.001 pc runs. If either statistic shifts by more than the run-to-run scatter, the headline results depend on the ad hoc softening; if they persist, Newtonian binary contamination is not the driver. Also record the pairwise separation distribution during the runs to quantify the time spent in the 0.001–0.024 pc Newtonian band.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central quantitative results rest on the claim (Abstract, Sec. 3.5) that softening with ε=0.001 pc suppresses the Newtonisation of compact subsystems. That claim is internally inconsistent. For a 0.5 Msun pair, the softened two-body force in Eq. (33) peaks at ≈0.385 Gm/ε² ≈ 800 pc/Myr², far above a0=3.8 pc/Myr²; the MLD/Newtonian transition for this pair occurs where Gm/r² ≈ a0, i.e. r≈0.024 pc, which is 24 times larger than ε. Consequently, any pair with separation in (0.001, 0.024) pc is internally Newtonian, and Sec. 4.2 demonstrates that such binaries follow Newtonian rather than MONDian galactic orbits. Over 1 Gyr of relaxation in a 400-particle cluster, close encounters in this separation range are frequent, so the simulated cluster is a hybrid of MOND-enhanced and Newtonian subsystems. The claimed leading-tail excess (q50-200≈1.5–2) and ~25% faster dissolution could therefore be caused by these Newtonian binaries or by the cluster recoil they induce (Sec. 5.2), rather than by 'general MONDian dynamics'. The choice ε=0.001 pc is thus not merely an approximation of an unknown field theory; it fails to realize the paper's own stated physics goal.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper postulates a vectorial Milgrom-law dynamics (MLD) for discrete N-body systems (Eq. 10), extends the standard Hermite integration scheme to compute MONDian accelerations and jerks, and uses it to simulate the tidal-tail evolution of a 400-particle, 200 Msun open cluster on a circular orbit at 8300 pc in a flat rotation curve. The central results are that in MLD the leading tidal tail is more populated than the trailing tail (q50-200 between 1.5 and 2) and that the cluster dissolves about 25% faster than in a Newtonian control run. The paper also presents analytic/numerical tests of an isolated binary, binaries in an external field, a hierarchical triple, and an isolated Plummer sphere, and compares the deep-MOND two-body force with Milgrom's field-theoretic expression.","tokens_in":19588,"tokens_out":5927,"duration_ms":61123,"significance":"If correct, the paper provides the first collisional direct N-body treatment of open clusters in a MOND-type dynamics, and it makes a falsifiable prediction that low-mass open clusters in the Solar neighborhood should show leading-tail excesses and enhanced dissolution relative to Newtonian expectations. The analytic derivation of the MONDian jerk (Eqs. 26-28) and the binary/triple sanity checks are internally consistent and clearly presented. The paper is also explicit about many caveats, including the lack of a variational principle for MLD and the approximate nature of the softening procedure. However, the significance is diminished by the ad hoc status of the central equation of motion and by the softening-scale inconsistency discussed below, because the quantitative claims may describe a hybrid Newtonian/MOND hybrid system rather than pure MOND dynamics.","major_comments":[{"comment":"The claim that softening with epsilon=0.001 pc suppresses the Newtonisation of compact subsystems is internally inconsistent. For a 0.5 Msun pair, Gm/r^2 = a0 at r = sqrt(Gm/a0) ≈ 0.024 pc, which is 24 times larger than epsilon. The softened force (Eq. 33) differs appreciably from the unsoftened force only for r < epsilon; in the range 0.001 < r < 0.024 pc the force is nearly Newtonian and exceeds a0. Thus the simulation contains internally Newtonian binaries, and Sec. 4.2 demonstrates that such binaries follow Newtonian rather than MONDian Galactic orbits. The paper provides no analysis of how often such binaries form in the cluster or how the tail asymmetry and dissolution rate depend on epsilon. The central quantitative result (q50-200 ≈ 1.5-2, 25% faster dissolution) could therefore be caused by these Newtonian binaries or by the recoil they induce (Sec. 5.2), rather than by the 'general MONDian dynamical concept' claimed in Sec. 6.","section":"Sec. 3.5 and Sec. 5.1, Eqs. (32)-(33), Fig. 13"},{"comment":"The equation of motion (Eq. 10) is a postulate that is not derived from AQUAL or QUMOND, and the paper correctly states that no variational principle is known. Given this, the conclusion in Sec. 6 that the tail asymmetry is 'a property of the general MONDian dynamical concept' because it appears in both QUMOND and MLD is too strong. MLD is an ad hoc prescription with an additional softening parameter, and the QUMOND simulations cited operate at different cluster masses (≳5000 Msun). At minimum, the abstract and conclusions should restrict the claim to 'MLD with the chosen softening' and state explicitly that it remains to be shown that actual MOND field theories produce the same effect for 200 Msun clusters.","section":"Sec. 2, Eq. (10), and Sec. 6"},{"comment":"The headline numbers are based on only five realizations per model. Fig. 20 shows large run-to-run scatter in q50-200, yet no uncertainties or significance tests are reported for the quoted range q50-200 = 1.5-2. Similarly, the remaining fractions after 1 Gyr are given as 0.26 ± 0.07 (MLD) and 0.41 ± 0.06 (Newtonian); the 1-sigma bands are separated by only about 1.5 sigma, making the claim of a ~25% faster dissolution statistically fragile. The paper should provide per-realization values, standard errors on the mean, and a test (e.g., a t-test or bootstrap) before drawing quantitative conclusions.","section":"Sec. 5.4 and Sec. 5.5, Figs. 19-21"},{"comment":"The Hermite scheme is used for a non-conservative, non-Hamiltonian system, but the paper gives no numerical convergence tests (e.g., energy or momentum diagnostics, or runs with shorter time steps) to verify that the integration error is controlled over 1 Gyr. Since the MLD equations of motion do not conserve the Newtonian integrals, the usual error checks based on energy conservation are unavailable, and the paper's Fig. 11/Fig. 21 results could depend on the integration accuracy. A convergence study for at least one MLD cluster run is needed to support the quantitative dissolution and asymmetry claims.","section":"Sec. 3.1-3.2 and Sec. 5.2"}],"minor_comments":[{"comment":"The units of a0 in the Newtonian limit are written as '10^-20 Myr/pc^2'; they should be pc/Myr^2 (or equivalent) to match the text and equations.","section":"Sec. 3.3"},{"comment":"The sentence 'the masses of the particles in the kinetic potential turn into their square roots' should read 'kinetic term' rather than 'kinetic potential'.","section":"Sec. 4.1"},{"comment":"Figure 13 would be much more informative if it included a vertical line at r ≈ 0.024 pc, the radius where a = a0 for a 0.5 Msun particle, because that is the scale at which the MOND/Newtonian transition occurs for the particles used in the cluster simulations.","section":"Fig. 13"},{"comment":"The citation 'Milgrom (2014, E.q 23)' contains a typo; it should be 'Eq. (23)'.","section":"Sec. 4.5"},{"comment":"In the phrase 'In lack of a known conserved quantity in MLD', 'lack' should be 'the absence'.","section":"Sec. 4.4"},{"comment":"The reference to 'solar neighborhood as used in related studies' should be followed by a period and likely the appropriate citation (e.g., Jerabkova et al. 2021).","section":"Sec. 5.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is honest about the exploratory nature of MLD, but the softening-scale problem is a genuine internal inconsistency that affects the interpretation of the main result. The fix is feasible in principle: the authors should re-run the cluster simulations with a larger softening parameter (e.g., 0.03 pc) and, importantly, quantify the frequency and orbital effects of near-Newtonian binaries that form in the cluster. They should also add statistical error estimates. The introductory discussion of the Bullet cluster is long and largely orthogonal to the paper's core; it could be shortened. If the authors can demonstrate that the asymmetry persists under these changes, the paper would make a valuable contribution to the MOND-vs-Newtonian debate at open-cluster scales."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know: this paper builds a direct N-body scheme for Milgrom-law dynamics (MLD) and applies it to a 200 Msun, 400-particle open cluster. The genuinely new pieces are the Hermite integration with MONDian jerks, an analytic deep-MOND two-body Lagrangian, and the specific quantitative claims: leading/trailing tail ratio q50-200 ≈ 1.5–2 and roughly 25% faster dissolution than Newtonian. The qualitative asymmetry was already reported in QUMOND simulations (Thomas et al. 2018; Kroupa et al. 2022), so the new output is the MLD tool itself and these numbers.\n\nThe paper is transparent about the status of Eq. (10): it is a postulate, not derived from AQUAL or QUMOND, and the author explicitly says no variational principle is known. The binary, triple, and Plummer-sphere tests are reasonable sanity checks. The jerk derivation (Eqs. 26–28) is internally consistent, and no parameter is fitted to the tail asymmetry or dissolution outputs; a0, vc, and the cluster parameters come from prior work.\n\nThe soft spot is load-bearing. The paper sets ε = 0.001 pc to \"suppress the Newtonisation of compact subsystems.\" For a 0.5 Msun pair, the MLD/Newtonian transition sits at r ≈ 0.024 pc (where Gm/r² = a0). So any pair with separation between 0.001 and 0.024 pc is internally Newtonian, not MONDian. Over 1 Gyr in a 400-particle cluster, such pairs form transiently, making the simulated cluster a hybrid of MOND-enhanced and Newtonian subsystems. The paper's own Sec. 4.2 shows that internally Newtonian binaries follow Newtonian galactic orbits. That means the tail asymmetry and dissolution rate could be influenced by close binaries and the associated recoil, not solely by \"general MONDian dynamics.\" The softening choice is thus not a harmless numerical detail; it fails to realize the paper's stated physics goal.\n\nOther concerns are smaller: only five MLD realizations, no error bars on q beyond the plotted spread, no code or seeds released, and the Sec. 6 conclusion that the asymmetry is \"a property of the general MONDian dynamical concept\" is too strong given that MLD is one postulated discretization and the softening issue is unresolved. The literature discussion is fine, and the stated limitations in the abstract and conclusions are honest.\n\nWho is this for? People modeling discrete stellar systems in MOND-like gravity. It deserves a serious referee because it opens a new numerical direction and makes falsifiable predictions. But the referee should push for convergence tests over ε, a demonstration that the signal survives when close binaries are treated consistently, and release of code and seeds. I would engage with it but not yet cite the quantitative results as robust.","headline":"A useful first MLD N-body tool with sensible sanity checks, but the headline tail asymmetry and dissolution numbers rest on an ad hoc softening that does not actually suppress Newtonisation of close pairs.","tokens_in":20091,"tokens_out":2509,"would_cite":false,"duration_ms":26186,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Applying Milgrom's law to discrete stars, this paper finds that low-mass open clusters develop leading tidal tails with up to twice as many stars as the trailing tail and dissolve about 25% faster than Newtonian clusters.","keywords":["MOND","modified Newtonian dynamics","Milgrom-law dynamics","open star clusters","tidal tails","N-body simulations","external field effect","cluster evaporation"],"falsifier":"Measure the 50\\,--\\,200 pc leading-to-trailing member ratio, $q_{50-200}$, and the member-loss fraction for a sample of nearby open clusters near 200 $M_\\odot$ with reliable astrometric membership; MLD predicts a mean $q_{50-200}$ between 1.5 and 2 and roughly 25% faster evaporation than Newtonian N-body models, so a sample that sits at $q\\approx 1$ with Newtonian-loss rates would refute the central claim.","tokens_in":19026,"feed_emoji":"⭐","tokens_out":8542,"duration_ms":82759,"temperature":0.7,"pith_summary":"This paper aims to extend MOND-style gravity down to the discrete, low-mass regime where real open star clusters live. Its route is Milgrom-law dynamics (MLD): Milgrom's law, $\\mu(|\\mathbf{a}|/a_0)\\mathbf{a}=\\mathbf{g}$, is asserted as a vector equation for every star, with a small softening length to keep compact subsystems from behaving Newtonianly. Simulating a 400-particle, 200-solar-mass cluster on a circular orbit at 8300 parsecs in a flat rotation curve, the paper finds the leading tidal tail carries up to twice as many stars as the trailing one and the cluster dissolves about 25% faster than its Newtonian counterpart, retaining 26% instead of 41% of its stars after 1 Gyr. Because the same asymmetry already appears in QUMOND field-theory simulations of heavier clusters, the paper concludes lopsided tidal tails are a generic MOND effect rather than an artifact of one formulation. Nearby open clusters could therefore offer a small-scale observational test of MOND.","feed_headline":"Simulated open clusters lose 25% more stars under Milgrom-law gravity","feed_subtitle":"A 200-solar-mass cluster keeps 26% of its stars after 1 Gyr, versus 41% in Newtonian gravity.","key_machinery":"The load-bearing object is the vectorial discrete Milgrom law, Eq. (10): $$\\mu\\left(\\frac{|\\mathbf{a}_i|}{a_0}\\right)\\!\\mathbf{a}_i = G\\sum_{j\\neq i} \\frac{m_j\\,(\\mathbf{r}_j-\\mathbf{r}_i)}{|\\mathbf{r}_j-\\mathbf{r}_i|^3}.$$ This replaces the AQUAL/QUMOND field equations, which need smooth densities and become impractical below about 5000 $M_\\odot$. The integration is a standard Hermite predictor-corrector scheme extended to supply both the MONDian acceleration and its jerk, with the standard interpolation function $\\mu(x)=x/\\sqrt{1+x^2}$. A softening length $\\varepsilon=0.001$ pc ($\\approx 206$ AU) is introduced both to regularize close encounters and, physically, to suppress the Newtonisation of compact subsystems, so the clusters stay internally in the MOND regime. For the isolated two-body deep-MOND case the paper derives an explicit Lagrangian whose conserved MLD momentum and centre of mass carry the conservation-law analysis.","core_discovery":"The central claim is that in Milgrom-law dynamics a low-mass open star cluster embedded in a Galactic disk develops a persistent leading-tail excess: in the 50\\,--\\,200 pc annulus the leading arm holds 1.5 to 2 times as many stars as the trailing arm, and the cluster loses its members about 25% faster than in Newtonian dynamics, with a mean retained fraction of 0.26 after 1 Gyr versus 0.41 in the Newtonian models. The same qualitative asymmetry had been found in QUMOND simulations of heavier clusters, and the paper argues this convergence means the asymmetry belongs to MOND generally, not to a particular equation. On the dynamical side, MLD does not conserve the Newtonian linear momentum, angular momentum, or Hamiltonian; for an isolated binary in the deep-MOND limit an alternative MLD centre of mass moves uniformly while the Newtonian centre of mass wobbles around it, and the MLD equations follow from a Lagrangian with a logarithmic potential and square-root masses.","pith_inferences":["If MLD is a fair stand-in for MOND, the observed Hyades tail asymmetry, cited as a 6.7$\\sigma$ outlier in Newtonian stochastic models, would be the expected MOND signature rather than a rare fluctuation; this is a consequence the paper points toward but does not itself simulate for the Hyades.","The 0.001 pc softening is doing physical work, effectively assuming no internally Newtonian binaries exist in the cluster; varying $\\varepsilon$ or seeding the cluster with hard binaries would be a direct numerical test of how the tail asymmetry and evaporation rate depend on that assumption.","Because the external field effect enters through the factor $\\mu(a_{\\rm ext}/a_0)$, the predicted asymmetry should depend on the cluster's galactic radius and orbital speed; comparing clusters at different radii in the same survey could separate MOND's tail asymmetry from bar- or spiral-arm-induced perturbations."],"forward_implications":["A 200 $M_\\odot$, 400-particle open cluster on a circular 8.3 kpc orbit retains 26% of its stars after 1 Gyr in MLD, versus 41% in Newtonian gravity.","The $q$-parameter $N_{\\rm lead}/N_{\\rm trail}$ in the 50\\,--\\,200 pc annulus settles between 1.5 and 2 in MLD, while Newtonian models stay near 1.","The leading-tail excess appears in both QUMOND field-theory simulations and discrete MLD simulations, which the paper reads as evidence that asymmetric tails are inherent to MOND-like dynamics.","Newtonian conservation laws fail in MLD, so quantities like the Newtonian centre of mass and angular momentum drift or oscillate instead of staying fixed.","Nearby open clusters, with their tails resolved by astrometry, become direct test beds for whether gravity below $a_0$ is Newtonian or MONDian."],"supporting_citations":[{"why":"Introduces the empirical acceleration law that MLD generalizes to discrete N-body systems.","marker":"Milgrom 1983a,b,c"},{"why":"Supplies the AQUAL field theory whose conservative structure MLD approximates and whose deep-MOND two-body force is compared in Sec. 4.5.","marker":"Bekenstein & Milgrom 1984"},{"why":"Identifies the linear-momentum non-conservation that arises when Milgrom's law is applied directly, the issue motivating the MLD formulation.","marker":"Felten 1984"},{"why":"Provides QUMOND, the quasi-linear field theory used in the higher-mass comparison simulations.","marker":"Milgrom 2010"},{"why":"Reports QUMOND simulations of Pal 5 showing asymmetric tidal tails, the qualitative result this work reproduces in MLD.","marker":"Thomas et al. 2018"},{"why":"Presents QUMOND simulations of a heavy open cluster and defines the $q$-parameter for tail asymmetry used here.","marker":"Kroupa et al. 2022"},{"why":"Provides the observed asymmetric tails in nearby open clusters that motivate simulating low-mass clusters in MOND.","marker":"Jerabkova et al. 2021"},{"why":"Defines the asymmetry and $q$ metrics and gives the stochastic Newtonian baseline for tail asymmetry.","marker":"Pflamm-Altenburg et al. 2023"},{"why":"Supplies the direct-summation Hermite integration scheme on which the MLD acceleration/jerk extension is built.","marker":"Aarseth 2003"},{"why":"Gives the $G^{-1/2}$ dissolution-time scaling used to interpret the faster MLD evaporation rate.","marker":"Baumgardt & Makino 2003"}],"fun_headline_variants":["Milgrom gravity doubles leading tail in open clusters","Simulated MOND clusters shed 25% more stars","MOND predicts asymmetric tidal tails in small clusters","Star cluster tails tip under Milgrom-law dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that Milgrom's law, applied star by star as a vector relation with a fixed 0.001 pc softening, is a valid approximation to real MOND for low-mass open clusters; if a full MOND field theory would not reduce to this vector law for point masses, the simulated tail asymmetry and the 25% faster dissolution would not describe actual clusters.","fun_headline_variants_meta":{"raw":{"variants":["Milgrom gravity doubles leading tail in open clusters","Simulated MOND clusters shed 25% more stars","MOND predicts asymmetric tidal tails in small clusters","Star cluster tails tip under Milgrom-law dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000253,"raw_usage":{"total_tokens":1634,"prompt_tokens":1084,"completion_tokens":550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":700,"completion_tokens_details":{"reasoning_tokens":487}},"tokens_in":700,"tokens_out":550,"duration_ms":5940,"temperature":1.0,"reasoning_tokens":487,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T16:00:45.199075+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the 50\\,--\\,200 pc leading-to-trailing member ratio, $q_{50-200}$, and the member-loss fraction for a sample of nearby open clusters near 200 $M_\\odot$ with reliable astrometric membership; MLD predicts a mean $q_{50-200}$ between 1.5 and 2 and roughly 25% faster evaporation than Newtonian N-body models, so a sample that sits at $q\\approx 1$ with Newtonian-loss rates would refute the central claim.","supporting_citations":[{"cited_title":"& Milgrom , M","cited_arxiv_id":null,"evidence_quote":"Supplies the AQUAL field theory whose conservative structure MLD approximates and whose deep-MOND two-body force is compared in Sec. 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the linear-momentum non-conservation that arises when Milgrom's law is applied directly, the issue motivating the MLD formulation."},{"cited_title":"2010, , 403, 886","cited_arxiv_id":null,"evidence_quote":"Provides QUMOND, the quasi-linear field theory used in the higher-mass comparison simulations."},{"cited_title":"F., Famaey , B., Ibata , R., et al","cited_arxiv_id":null,"evidence_quote":"Reports QUMOND simulations of Pal 5 showing asymmetric tidal tails, the qualitative result this work reproduces in MLD."},{"cited_title":"2022, , 517, 3613","cited_arxiv_id":null,"evidence_quote":"Presents QUMOND simulations of a heavy open cluster and defines the $q$-parameter for tail asymmetry used here."},{"cited_title":"2023, , 671, A88","cited_arxiv_id":null,"evidence_quote":"Defines the asymmetry and $q$ metrics and gives the stochastic Newtonian baseline for tail asymmetry."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the direct-summation Hermite integration scheme on which the MLD acceleration/jerk extension is built."},{"cited_title":"& Makino , J","cited_arxiv_id":null,"evidence_quote":"Gives the $G^{-1/2}$ dissolution-time scaling used to interpret the faster MLD evaporation rate."}],"review_version":1}